Daniel Finley (Professor)

Ph.D. 1968, University of California at Berkeley

Brief Description of Research

My current work involves exact descriptions of solutions of the vacuum Einstein field equations, for algebraically-degenerate Petrov types, especially those related to non-trivial (twisting) gravitational waves.
The complexity of the gravitational equations has led me to interests in other sorts of nonlinear physics and in general methods for studying them, the most common being infinite-dimensional Lie algebras of local and/or non-local symmetries of the associated nonlinear PDE's.

The main interest is in solutions to Einstein's field equations, and in perturbations of both the solutions themselves and massless field equations defined over the corresponding curved spaces. The approach to solutions is physical in motivation but mathematical in content. I often use the methods of complex differential manifolds, leading to the study of the structure of all possible self-dual spaces (H-spaces) with complex or Euclidean signature. A natural generalization of this is the study of HH-spaces, which may possess Minkowski signature, real, algebraically-degenerate cross-sections. Work on HH-spaces is proceeding by division into distinct Petrov types, leading particularly to the study of gravitational waves. Before his death much of this work was done in active collaboration with Jerzy Plebanski at the Instituto de Investigaciones y Estudios Avanzados del Instituto Politecnico Nacional in Mexico City and the University of Warsaw in Poland.

Some publications in this area are the following:

Areas of investigation in nonlinear physics include methods of finding families of particular, exact solutions of many different systems of nonlinear equations, motivation for the choices being found in problems in hydrodynamic flow, chemical mixing, nonlinear optics, three-level laser systems, and gravitational waves. These methods are used to generate families of soliton or breather solutions or Bäcklund transformations between equations, and to establish connections with many (other) interesting mathematical structures, including complex manifold theory, twistor methods, infinite solvable algebras, Kac-Moody algebras, and larger algebras (of infinite growth). John K. McIver is a collaborator in this work.

Some publications in this area are the following:

Back to
Daniel Finley's Dept. Page
Daniel Finley's Main Links Page

Last updated/modified: 30 May, 2013

finley@unm.edu, finley@phys.unm.edu